Octahedral pyramid

The regular 16-cell has octahedral pyramids around every vertex, with the octahedron passing through the center of the 16-cell. Therefore placing two regular octahedral pyramids base to base constructs a 16-cell. The 16-cell tessellates 4-dimensional space as the 16-cell honeycomb.

Exactly 24 regular octahedral pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid). This construction yields a 24-cell with octahedral bounding cells, surrounding a central vertex with 24 edge-length long radii. The 4-dimensional content of a unit-edge-length 24-cell is 2, so the content of the regular octahedral pyramid is 1/12. The 24-cell tessellates 4-dimensional space as the 24-cell honeycomb.

The octahedral pyramid is the vertex figure for a truncated 5-orthoplex, {{CDD|node_1|3|node_1|3|node|3|node|4|node}}.

:160px

The graph of the octahedral pyramid is the only possible minimal counterexample to Negami's conjecture, that the connected graphs with planar covers are themselves projective-planar.{{citation

| last = Hliněný | first = Petr

| doi = 10.1007/s00373-010-0934-9

| issue = 4

| journal = Graphs and Combinatorics

| mr = 2669457

| pages = 525–536

| title = 20 years of Negami's planar cover conjecture

| url = http://www.fi.muni.cz/~hlineny/papers/plcover20-gc.pdf

| volume = 26

| year = 2010| citeseerx = 10.1.1.605.4932

| s2cid = 121645

}}

Example 4-dimensional coordinates, 6 points in first 3 coordinates for cube and 4th dimension for the apex.

$$\backslash begin\{array\}\{lllr\}$$

(\pm 1, & 0, & 0; & 0) \\

(0, & \pm 1, & 0; & 0) \\

(0, & 0, & \pm1; & 0) \\

(0, & 0, & 0; & \ 1)

\end{array}

The dual to the octahedral pyramid is a cubic pyramid, seen as a cubic base and 6 square pyramids meeting at an apex.

:160px

Example 4-dimensional coordinates, 8 points in first 3 coordinates for cube and 4th dimension for the apex.

$$\backslash begin\{array\}\{lllr\}$$

(\pm 1, & \pm 1, & \pm 1; & 0) \\

(0, & 0, & 0; & 1)

\end{array}

{{-}}

{{Infobox 4-polytope

| Name = Square-pyramidal pyramid

| Image_File = Square pyramid pyramid.png

| Image_Caption =

| Type = Polyhedral pyramid

| Schläfli = {{nowrap|( ) ∨ [( ) ∨ {4}] }}
{{nowrap|1=[( )∨( )] ∨ {4} = { } ∨ {4} }}
{{nowrap|{ } ∨ [{ } × { }]
{ } ∨ [{ } + { }] }}

| CD =

| Cell_List = 2 ( )∨{4} 30px
4 ( )∨{3} 30px

| Face_List = 12 {3}
1 {4}

| Edge_Count = 13

| Vertex_Count = 6

| Vertex_Figure =

| Petrie_Polygon =

| Coxeter_Group =

| Symmetry_Group = [4,1,1], order 8
[4,2,1], order 16
[2,2,1], order 8

| Dual = Self-dual

| Property_List = convex, regular-faced

| Index =

}}

File:Square pyramid pyramid edgecenter.png

The square-pyramidal pyramid, {{nowrap|( ) ∨ [( ) ∨ {4}]}}, is a bisected octahedral pyramid. It has a square pyramid base, and 4 tetrahedrons along with another one more square pyramid meeting at the apex. It can also be seen in an edge-centered projection as a square bipyramid with four tetrahedra wrapped around the common edge. If the height of the two apexes are the same, it can be given a higher symmetry name {{nowrap|1=[( ) ∨ ( )] ∨ {4} = { } ∨ {4},}} joining an edge to a perpendicular square. {{KlitzingPolytopes|..//incmats/squasc.htm|Segmentotope|squasc, K-4.4}}

The square-pyramidal pyramid can be distorted into a rectangular-pyramidal pyramid, {{nowrap|{ } ∨ [{ } × { }] }} or a rhombic-pyramidal pyramid, {{nowrap|{ } ∨ [{ } + { }],}} or other lower symmetry forms.

The square-pyramidal pyramid exists as a vertex figure in uniform polytopes of the form {{CDD|node|p|node_1|q|node_1|r|node|4|node}}, including the bitruncated 5-orthoplex and bitruncated tesseractic honeycomb.

:120px120px

Example 4-dimensional coordinates, 2 coordinates for square, and axial points for pyramidal points.

$$\backslash begin\{array\}\{lllr\}$$

(\pm 1, & \pm 1; & 0; & 0) \\

(0, & 0; & 1; & 0) \\

(0, & 0; & 0; & \ \ 1)

\end{array}

{{reflist}}

• {{GlossaryForHyperspace |anchor=Pyramid |title=Pyramid}}
• {{KlitzingPolytopes|../explain/segmentochora.htm|4D|Segmentotopes}}
• {{KlitzingPolytopes|..//incmats/octpy.htm|Segmentotope|octpy, K-4.3}}
• Richard Klitzing, [http://bendwavy.org/klitzing/pdf/edge-facetings_color.pdf Axial-Symmetrical Edge Facetings of Uniform Polyhedra]

Category:4-polytopes

Category:Pyramids (geometry)

{{Polychora-stub}}